There's a small bakery/cafe on main street.
Park East delivers on Thursdays. You can order out everything for the weekend.
One of the things they deliver is the luncheon/kiddush for the Hamptons Synagogue. Don't plan lunch, just stay for kiddush. Ditto for seudah shlishi. On Friday evening, if you wish you can reserve a paid space at the dinner. There will be a speaker worth listening to at the Friday night dinner.
There are some small inns in the center where people stay, often when thinking of purchasing for next season. Or visiting friends with not quite enough bedrooms. They make refrigerator space available to guests and leave a Shabbos kettle on for tea.